2

which can be derived with unit impulse. For practical structural engineering problems, the damping factors are usually small (p < 15%). p* and p* may be replaced by p and p, respectively. Thus, Equation 3.24a becomes

When the structure is subjected to ground acceleration xg, the motion equation is

The displacement response is similar to Equation 3.24b in which F(D) is replaced by Mxg. To find the maximum displacement over the entire or partial record of the earthquake, the integration must be carried for that record in order to determine the largest displacement corresponding to a given frequency and damping factor. The integration is shown in Equation 3.26 with consideration of the nature of ground motion, the negative sign has no real significance and can be ignored:

from which the displacement response spectrum can be established. Spectra for velocities, Sv and accelerations, Sa, are then obtained from

Response spectra computed for the N-S component of El Centro Earthquake, May 18, 1940, are given in a tripartite logarithmic plot as shown in Figure 3.9. Note that when the frequency is large, the relative displacement is small and the acceleration is large, but when the frequency is small, the displacement is large and the acceleration is relatively small; the velocity is always large around the region of intermediate frequencies. Since the response does not reflect the real time-history response but a maximum value, the response is called pseudo-response such as pseudo-displacement, pseudo-velocity, and pseudo-acceleration. Note that curves are jagged at different frequencies due to randomness of seismic input.

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